126 7 A Primer in Stochastic Integration
On the other hand we have on{T> 1 }and fort≥ 1
(H·A)t=∫t∧T0du
1 −u=+∞.
The next theorem gives a necessary and sufficient condition for a stochastic
integral of a local martingale to be again a local martingale. The result in this
form is due to Ansel and Stricker [AS 94]. An earlier form was given byEmery ́
[E 79].
Theorem 7.3.7 (Ansel and Stricker).LetMbe anRd-valued local mar-
tingale, letHbeRd-valued and predictable. LetHbeM-integrable in the sense
of semi-martingales. ThenH·Mis a local martingale if and only if there is
an increasing sequence of stopping timesTn↗∞as well as a sequence of
integrable functionsθn≤ 0 , such that(H,∆M)Tn≥θn.
Some explanation on the notation seems in order: the process ∆M =
(∆Mt)t≥ 0 is the process formed by the jumps ofM; it is different from zero
only at the jumps of the cadlag processMwhere the formula (∆M)t(ω)=
Mt(ω)−Mt−(ω) holds. Hence the process (H,∆M)=(Ht,(∆M)t)t≥ 0 is the
process of jumps ofH·M. The condition (H,∆M)Tn≥θnmeans that, for
a.e.ω∈Ω, the jumps of the process ((H·M)t)t≥ 0 such that 0≤t≤Tn(ω)
all are bounded from below byθn(ω)almostsurely.
Proof of Theorem 7.3.7.We first prove necessity. IfH·Mis a local martingale
then it is locally inH^1. Hence there is an increasing sequence of stopping
timesTn↗∞such that supt≤Tn|(H·M)t|∈L^1. Hence|(H,∆M)Tn|≤
2supt≤Tn|(H·M)t|∈L^1.
The sufficiency of the hypothesis is less trivial. We will show that, for each
n∈N, the process (H·M)Tnis a local martingale and we may therefore drop
the stopping timesTnand replaceθnbyθ.Let
Ut=∑
s≤t(^1) {|∆Ms|≥ 1 or(Hs,∆Ms)≥ 1 }∆Ms
BecauseMandH·Mare semi-martingales, their jumps of high magnitude
(here≥1) form a discrete set, i.e., such that its intersection with each compact
subset of [0,∞[ is finite for a.e.ω∈Ω. The processUis therefore a cadlag,
adapted process of finite variation and hence a semi-martingale. AlsoHis
U-integrable. Indeed, every (Rd-valued, predictable) process isU-integrable.
Let nowY=M−U.SinceHisM-andU-integrable it is alsoY-integrable.
The semi-martingaleY has jumps of magnitude≤1 and hence is a special
semi-martingale. Its canonical decomposition is denoted asY=N+B.Let
V=B+U=M−N, the processVis the difference of two local martingales
and is therefore a local martingale, moreover it has paths of locally bounded
variation as it is the sum of the processBandU. BecauseH·Y=H·M−H·U
has bounded jumps it is also special and by Lemma 7.3.5H·Bexists as an
ordinary integral. ThereforeH·V exists as an ordinary Lebesgue-Stieltjes
integral too. We have to show two things